An open set of maps for which every point is absolutely nonshadowable

AN OPEN SET OF MAPS FOR WHICH EVERY POINTIS ABSOLUTELY NONSHADOWABLEGUO-CHENG YUAN AND JAMES A. YORKEAbstract. We consider a class of nonhyperbolic systems, for whichthere are two �xed points in an attractor having a dense trajectory;the unstable manifold of one has dimension one and the other's istwo dimensional. Under the condition that there exists a direc-tion which is more expanding than other directions, we show thatsuch attractors are nonshadowable. Using this theorem, we provethat there is an open set of di�eomorphisms (in the Cr-topology,r > 1) for which every point is absolutely nonshadowable, i. e. ,there exists � > 0 such that for every � > 0, almost every �-pseudotrajectory starting from this point is �-nonshadowable.1. IntroductionIn study of chaotic systems, a researcher is often obliged to rely onnumerical simulations because direct analytical methods are not avail-able. To ensure their validity, it is crucial that numerical trajectoriesstay close to, in other words, they are shadowed (see Section 2 forde�nition) by, true trajectories; otherwise, the meaning of numericalresults is far from obvious. Although compact hyperbolic invariantsets are shadowable as proved by Anosov [1] and Bowen [2], virtuallyall chaotic attractors that scientists encounter are nonhyperbolic. Forexample, the \H�enon" strange attractors constructed by Benedicks and1991 Mathematics Subject Classi�cation. Primary h58F13i; Secondary h58F12,58F14, 58F15i.This research was supported by the National Science Foundation and Depart-ment of Energy. 1

2 GUO-CHENG YUAN AND JAMES A. YORKECarleson [3] are not hyperbolic: the angle between stable and unsta-ble manifolds is not bounded away from zero. There seems to be afeeling among many mathematicians that nonhyperbolicity generallyimplies nonshadowability, but there are counterexamples. Nonhyper-bolic attractors can have the shadowing property. Let F� be a oneparameter family of maps or di�eomorphisms with a periodic attractorthat undergoes a cascade of period doublings, limiting on the so-called\Feigenbaum" parameter �c (which was �rst described by Myrberg [4]).The corresponding \Feigenbaum" attractor at this value is not hyper-bolic, but it does satisfy the shadowing property. Furthermore, if theabove example lies on a space M1, and on another space M2 there isa hyperbolic attractor, then the attractor for the product system ischaotic and has the shadowing property, but the attractor is not hy-perbolic. For a simpler example, Coven, Kan and Yorke [5] have shownthat tent maps are shadowable for all parameter values (i. e. , the ab-solute value of the slopes) between 1 and 2 except for a set having zeroLebesgue measure. (Indeed, this paper discusses tent maps with slopesbetweenp2 and 2, though it is trivially extendible to slopes between 1and 2. They also show that the set of parameters for which the mapis nonshadowable is uncountable.) In their case, the attractor is thewhole space. Similarly, in the set of parameter values where the logisticmap has a chaotic attractor there is a dense, uncountable set for whichit is shadowable and a dense uncountable set for which it is not. Inthe window of parameter values with an attracting periodic orbit, it isshadowable, see Smale and Williams [6] for the period 3 case, wherethey show the invariant set of points not in the period 3 basin of theattractor is hyperbolic.

NONSHADOWABLE SYSTEMS 3Several papers [7, 8, 9, 10, 11, 12, 13, 14] have given methods torigorously verify whether numerical trajectories are shadowed by truetrajectories. Results obtained from applying such methods to nonhy-perbolic maps on one and two dimensional phase spaces [7, 8, 9] suggestthat such systems are nonshadowable. The critical point of the logis-tic map is analogous to points of the chaotic attractors of H�enon mapswhere the stable and unstable sets are tangent. However, the H�enon at-tractor appears to have uncountably many such points, so the analysisis more complicated and the shadowing properties are yet unresolved.Our goal is to discuss another mechanism for nonshadowability whichwe call \dimension variability", i. e. , an attractor (with a densetrajectory) has at least two hyperbolic periodic orbits whose unsta-ble manifolds have di�erent dimensions. Using this phenomenon, wepresent an open set of maps (or di�eomorphisms) for which every pointis absolutely nonshadowable (see Section 2 for de�nition). In the nextsection we will analyze geometrically how dimension variability is anobstacle for shadowing. Some of our ideas have been sketched in Poonet. al [15].Dimension variability is related to studies of �nite shadowing time| i. e. , starting from an initial point, for how long can the pseudotrajectory be shadowed by a true trajectory (the shadowing distance iscomparable to the size of the attractor) | for nonhyperbolic systems.It has been conjectured [7, 8] that for typical dissipative maps beingiterated with numerical accuracy �, the average shadowing time is ofthe order 1=p�. This conjecture is supported by numerical studies ofthe logistic map and H�enon map. Recent studies discuss a new typeof map in which almost every trajectory has the following property.The Lyapunov exponents are non-zero, but there are arbitrarily long

4 GUO-CHENG YUAN AND JAMES A. YORKEsegments of the trajectory for which the number of the approximate\�nite-time" Lyapunov exponents that are positive is di�erent fromthe number of actual positive exponents. For such maps, the averageshadowing time can be much shorter, meaning that the shadowingdi�culty is more serious than it is for the logistic map and H�enon map.Systems with dimension variability have this property, because almostevery trajectory has arbitrarily long segments in which it remains neareach of the periodic orbits.2. Main TheoremLetM be an m-dimensional Riemannian manifold and f:M !M bea C1 map. Let A be a compact invariant set, we say A is Lyapunovstable if there exists a family of neighborhoods Ui of A such thatTi Ui = A and f(Ui) � Ui for all i. We say A is an attractor, if i) thereexists an neighborhood U such that for every x 2 U , the positive limitset !(x) � A; ii) A is Lyapunov stable; and iii) there exists p 2M suchthat !(p) = A. If A is an attractor, we say the set fp 2M : !(p) � Agis the basin of A. Notice that the usual de�nition of an attractor onlyrequires the �rst condition, but then it would include the pathologicalcases where nearby orbits have many intermediate iterates that stayaway from the invariant set. We exclude such pathological cases byadding condition ii) in our de�nition. We also add condition iii) toavoid unnecessarily large attractors. For example, the whole manifoldis an attractor under the usual de�nition, whereas under our de�nitionit is an attractor only if there exists an orbit that is dense everywherein the manifold.Given �; � > 0, we say �x = (xi)bi=a is a �-pseudo trajectory ifd(f(xi); xi+1) < � whenever a � i < b. Let x;� be the set of �-pseudo

NONSHADOWABLE SYSTEMS 5trajectories that start from x. If we consider xi+1 as being chosenat random from a uniform distribution in B(x; �) (from now on, weuse B(x; �) resp. B(V; �) to denote the �-neighborhood of a point xresp. a subset V of M), then x;� forms a Markov chain with each�-pseudo trajectory in x;� as a sample sequence [16]. We say \almostevery �-pseudo trajectory" has a property P if the event P occurs withprobability 1 for this Markov chain. Similar ideas have been discussedby Young [17] and Kifer [18].Given > 0, we say a pseudo orbit (xi)1i=0 comes within of a�nite pseudo orbit (yi)ni=0 if there exists ` � 0 such that d(x`+i; yi) < ,for 1 � i � n. We have the following observation.Proposition 2.1. Let A be an attractor and U be the basin of A. Letx 2 U . Assume that � > 0 is su�ciently small that there exists acompact neighborhood V � U of A such that x 2 V and f(B(V; �)) �V . Let �y := (y0; : : : ; yn) be a �-pseudo trajectory in B(A; �). Then foreach > 0, almost every �x := (xi)1i=0 2 x;� comes within of �y.Proof. Denote by Ex the event that a pseudo orbit �x 2 x;� comeswithin of �y. For each z 2 V , there exist an open neighborhood Uzof z, a positive number �z, and a positive integer nz such that foreach point z0 2 Uz, the probability for a �nite �-pseudo trajectory(zi)nzi=0 with z0 = z0 to come within of �y is greater than �z. SinceV is compact, there exists a �nite set fz(1); : : : ; z(k)g � V such thatSki=1 Uz(i) � V . This implies that there exist �� > 0 and n� > 0 suchthat for every z 2 V , the probability for a �-pseudo trajectory (zi)n�i=0with z0 = z to come within of �y is greater than ��. Note that if x 2 Vand �x 2 x;�, then fxig1i=0 � V . Each of the �nite �-pseudo trajectories(xi)jn�+n��1i=jn� , 0 � j <1, comes within of �y with probability greater

6 GUO-CHENG YUAN AND JAMES A. YORKEthan ��. Since there are in�nitely many independent trials, P (Ex) = 1.2 A �-pseudo trajectory �x := (xi)bi=a is �-shadowed by y if d(xi; f i(y)) <� when a � i � b. We say a point x is absolutely nonshadowable ifthere exists � > 0 such that for every � > 0, almost every �-pseudo tra-jectory (xi)1i=0 2 x;� is �-nonshadowable, i. e. , it cannot be �-shadowedby any true trajectory. Of course, x;� includes true trajectories, andthose are trivially shadowable; hence we can only require that almostevery �-pseudo trajectory is �-nonshadowable.Before we state our theorem, we need to introduce some notions,which play an important role in the proof of the well-known Hadamard-Perron Theorem [19]. (An excellent presentation of the proof can befound in [20], pages 242-257.) Let Sk = f(u; v) 2 Rk � Rm�k : juj �jvjg. We say Sk is the standard k-cone in Rm. A set C0 � Rm issaid to be a k-cone if C0 is isomorphic to Sk up to a linear change ofvariables. Let C � TM . We write Cx := C TTxM . Assume for x 2 A,Cx is nonempty and x 7�! Cx is continuous on A in the Hausdor�metric. We say C is a k-cone �eld on A if for each x 2 A, Cx is ak-cone. We say C is positively invariant if (Df)Cx � intCf(x) Sf0gfor all x 2 A.We say an invariant set A has dimension variability if there existhyperbolic periodic points p1; p2 in A, such that 0 < dim(W u(p1)) <dim(W u(p2)). Below we assume for convenience that these orbits are�xed points.Theorem 2.2. Let f :M ! M be a C1 map. Assume f satis�es thefollowing properties:1. f has an attractor A, and U is its basin;

NONSHADOWABLE SYSTEMS 72. (dimension variability) there exist hyperbolic �xed points p1; p2 inA, such that 0 < dim(W u(p1)) < dim(W u(p2));3. W u(p1) = A;4. write k := dim(W u(p1)); there exists a positively invariant k-cone�eld C on A.Then every point in U is absolutely nonshadowable.Remark 2.3. If f is a di�eomorphism, then the following condition isequivalent to Condition 4 in Theorem 2.2:4'. There exists a continuous splitting TAM = E1 � E2, wheredim(E1) = dim(W u(p1)), and positive constants K; � and � with� < �, such that: i) Df(Ei) = Ei; i = 1; 2; ii) for all v 2 E1 andn � 0, jDf�n(v)j � K��njvj; and iii) for all v 2 E2 and n � 0,jDfn(v)j � K�njvj. Note in particular � need not be less than 1.Remark 2.4. We have cited literature discussing shadowing failure dueto tangencies of stable and unstable manifolds. Condition 4 guaranteesthere are no tangencies. Hence dimension variability is a distinct mech-anism. In particular, Condition 4 prevents W u(p1) from being tangentto W s(p2) at any point in W u(p1)TW s(p2). If f is a di�eomorphism,then at each x 2 W u(p1)TW s(p2), W u(p1) is tangent to E1(x), andW s(p2) is tangent to a subspace of E2(x). Since the angle between E1and E2 is bounded away from 0, W u(p1) can not be tangent to W s(p2)at any point.Remark 2.5. Condition 4 is also a uniform condition to make it possiblefor us to obtain a proof. We do not know if the result is true withoutCondition 4.

8 GUO-CHENG YUAN AND JAMES A. YORKETo simplify notation, we prove Theorem 2.2 for the case dim(W u(p1))= 1 and dim(W u(p2)) = 2. The proof for the other cases follows in thesame manner.For the rest of this section, we assume all the conditions in Theo-rem 2.2 are satis�ed. We need some notation before proving the the-orem. For i = 1; 2, write Eu(pi) � TpiM for the unstable space (i. e. ,with respect to Df) at pi, and W u(pi) for the unstable manifold of pi.For � > 0, let W u� (p1) be the local unstable manifold of p1. For x 2 A,write C(x; �) := fu 2 Cx : juj � �g, and let expx : TxM ! M be theexponential map induced by the Riemannian metric. If � is su�cientlysmall, then expx maps C(x; �) di�eomorphically to a subset expxC(x; �)in M . For convenience, we do not distinguish between expxC(x; �) andC(x; �) in our notation.The following lemmas are needed in the proof of Theorem 2.2.Lemma 2.6. (a) There exists �0 > 0 such that if x 2 A, thenf(C(x; �))\B(f(x); �) � int(C(f(x); �))[ff(x)g;for each 0 < � < �0.(b) Moreover, there exists � > 0, such that if x 2 A, w 2 B(f(x); �),and w 62 intC(f(x); �), then d(w; f(C(x; �))) � �d(w; f(x)).Proof. Given x 2 A, � > 0, and v 2 TxM , jvj = 1, let l(x; v; �) =fexpx(tv) : 0 � t � �g. If v 2 Cx, then Dfxv 2 intCf(x). Thereforethere exists �x;v > 0 such that f(l(x; v; �))TB(f(x); �) � intC(f(x); �)Sff(x)g for 0 < � < �x;v. By continuity, there exists �x > 0 such thatfor all v 2 Cx with jvj = 1, f(l(x; v; �))TB(f(x); �) � intC(f(x); �)Sff(x)gfor 0 < � < �x. Also by continuity, there exists �0 such that for all x 2 Aand 0 < � < �0, f(C(x; �))TB(f(x); �) � intC(f(x); �)Sff(x)g.

NONSHADOWABLE SYSTEMS 9In the following we prove the second part of the lemma. SinceDfxCx � intCf(x) Sf0f(x)g and Dfx is a linear map, there exists such that if v 2 Tf(x)M and v 62 intCf(x) then d(v;DfxCx) � jvj.Since expx is a local di�eomorphism, the previous argument impliesstatement (b). 2Lemma 2.7. There exist �0; � > 0 and � > 1 with the following prop-erties.(a): On B(p2; �0) there exists an invariant two dimensional C0-(unstable)foliation such that if y, z are in the same leaf and ff i(y); f i(z)gni=0 �B(p2; �0), then d(f i(y); f i(z)) � ��id(y; z)for 0 � i � n.(b): On B(p1; �0) there exists an invariant (m-1)-dimensional C0-(stable) foliation such that if y, z are in the same leaf and ff i(y); f i(z)gni=0 �B(p1; �0), then d(f i(y); f i(z)) � (��i)�1d(y; z)for 0 � i � n.Proof. We only prove the �rst part. Similar arguments apply to thesecond part.If f is a linear map, then we de�ne the leaf through each x 2 B(p2; �0)by x +W u(p2). The foliation thus de�ned satis�es property (a).In general, f is locally conjugate to a linear map. Note that property(a) is preserved by conjugacies. The proof is complete. 2The foliations given by Lemma 2.7 are not unique. From now on,we �x one unstable foliation on B(p2; �0) and one stable foliation on

10 GUO-CHENG YUAN AND JAMES A. YORKEB(p1; �0). We write W u(y; p2; �0) for the leaf of this unstable folia-tion through y 2 B(p2; �0) and W s(z; p1; �0) for the leaf of this stablefoliation through z 2 B(p1; �0).Lemma 2.8. There exists �0 > 0 such that the following statement istrue for each 0 < � < �0 and � > 0.Let n 2 N and x 2 B(p2; �) be such that x; f(x); : : : ; fn(x) 2B(p2; �). Then there exists w 2 W u(x; p2; �) such that i) d(x; w) � �;and ii) f i(w) 62 C(f i(x); �), for 0 � i � n. Furthermore, if � is su�-ciently small, then w can be chosen so that d(x; w) � �=2.Proof. There exists �0 > 0 such that for each x 2 M , f maps B(x; �0)di�eomorphically onto its image f(B(x; �0)). Let x; f(x); : : : ; fn(x) 2ATB(p2; �0). Write W := Tni=0 f�i(B(f i(x); �0)). Then W containsa neighborhood of x. Therefore fn(W ) contains a neighborhood offn(x). In particular, there exists a curve l � W passing through fn(x)such that l � W u(fn(x); p2; �) and lTC(fn(x); �0) = ffn(x)g. Let l0 =f�n(l)TW . Then l0 is a curve passing through x and l0 � W u(x; p2; �).For each w 2 l0 di�erent from x, f i(w)(0 � i � n) is not contained inC(f i(x); �). In particular, w can be chosen so that d(w; x) � �. If � issu�ciently small, then � can be chosen so that d(w; x) � �=2. 2We say a �-pseudo trajectory �x is elementary if there exists i0, suchthat f(xi) = xi+1 for all i except for i = i0. The next lemma is the keystep toward the �nal proof of Theorem 2.2.Lemma 2.9. There exist �0 > 0 and n0 > 0 such that for all 0 <�0 < �0 and � > 0, there exist x 2 W u�0(p1) and an elementary �-pseudotrajectory �x = (xi)n0i=0, where x0 = x, such that �x can not be �0-shadowedby any y 2 C(x; �0).

NONSHADOWABLE SYSTEMS 11Proof. Let �0 be a small number such that the next two conditions aresatis�ed:1. Lemmas 2.6- 2.8 hold.2. f restricted to B(p1; �0) is one-to-one, and W u�0(p1) � f(W u�0(p1)).Fix �0 2 (0; �0). Given � > 0, there exists n1, such that ����n1 > 2�0,where � is as in Lemma 2.6 and � and � are as in Lemma 2.7. Let s > 0be su�ciently small, such that Sn1i=0 f i(B(p2; s)) � B(p2; �0). Choosex0 2 W u(p1), such that f(x0) 2 B(p2; s).In the following we will introduce a small perturbation at f(x0) suchthat the perturbed trajectory is not shadowable. Note that W u(p2) istwo dimensional whereas W u(p1) is only one dimensional. When thetrajectory is perturbed away fromW u(p1), the trajectory diverges fromW u(p1). We will see this leads to nonshadowability.We �rst prove that there exists a �-pseudo trajectory starting from x0which can not be �0-shadowed by any y 2 C(x0; �0). By Lemma 2.8, forevery � > 0, there exists w 2 W u(f(x0); p2; �0) such that i) d(f(x0); w) ��; and ii) f i(w) 62 C(f i+1(x0); �0) for 0 � i � n1. For the purpose ofshadowing, we let d(f(x0); w) � �=2. Therefored(fn1(w); fn1+1(C(x0; �0))) � d(fn1(w); f(C(fn1(x0); �0)))� �d(fn1w; fn1+1(x0))� ���n1d(w; f(x0))� ���n1�2> �0:In the above estimation, we have used Lemmas 2.6- 2.8.

12 GUO-CHENG YUAN AND JAMES A. YORKEThe elementary �-pseudo trajectory (x0; w; f(w); � � � ; fn1(w)) cannot be �0-shadowed by any y 2 C(x0; �0).If x0 2 W u�0 (p1), then by letting x equal x0 we are done. Otherwise,there exists x 2 W u�0 (p1) and n2 > 0, such that x0 = fn2(x). De�nexi = f i(x) for 0 � i � n2, and xi = f i�n2�1(w) for i � n2 + 1. Letn0 = n1 + n2 + 1, then �x = (xi)n0i=0 cannot be �0-shadowed by anyy 2 C(x; �0). 2Let �0, �0, x, �x and n0 be as in Lemma 2.9. For i < 0, let xi bethe unique point in W u�0(p1)T f�1(xi+1). De�ne �xi := (xj)n0j=i. Sincex 2 W u(p1), xi ! p1 as i! �1. Note that for each i < 0, �xi cannotbe �0-shadowed by any y 2 C(xi; �0).The next two lemmas extend the result in Lemma 2.9. They areneeded to �nish the proof.Lemma 2.10. Let �0, x, �x and n0 be as in Lemma 2.9. Then thereexists � > 0, such that �x is not �-shadowed by any y 2 B(C(x; �); �),for 0 < � < �02Proof. Fix �0 2 (0; �0). Let � > 0 be su�ciently small such thatmax1�i�n0 d(f i(y); f i(z)) < �0=2 whenever d(y; z) < �. If y 2 B(C(x; �0); �),then there exists z 2 C(x; �0) such that d(y; z) < �. From Lemma 2.9,d(fn0(z); xn0) > �0. Therefored(fn0(y); xn0) � d(fn0(z); xn0)� d(fn0(y); fn0(z)) > �0 � �02 = �02 :�x is not �0=2-shadowed by y. Let � = �0=2, then the proof is complete.2Lemma 2.11. De�ne �xi, i < 0 as in the paragraph that follows Lemma 2.9.Let �0 and � be as in Lemma 2.10. Then there exists n3 > 0, such that�x�n3 cannot be �-shadowed by any y 2M .

NONSHADOWABLE SYSTEMS 13Proof. Given y 2 B(p1; �) and n > 0 such that ff i(y)gni=0 � B(p1; �0),there exists z 2 W s(y; p1; �)TW u� (p1) such that ff i(z)gni=0 � B(p1; �0).By Lemma 2.7, d(f i(y); f i(z)) � (��i)�1d(y; z) for 0 � i � n. There-fore d(f i(y);W u� (p1)) � (��i)�1�.Let y 2 M and n3 � 1 be such that d(f i(y); xi�n3) < � for 0 � i �n3. Note that x 2 W u� (p1). From previous arguments, d(fn3(y); C(x; �))� d(fn3(y);W u� (p1)) � (��n3)�1 ��. We can choose n3 to be su�cientlylarge that (��n3)�1�� < �. Apply Lemma 2.10, then �x is not �-shadowedby fn3(y). Therefore �x�n3 is not �-shadowed by y. 2Proof of Theorem 2.2. In the previous lemmas, we have proved thatthe �-pseudo trajectory �x�n3 = (x�n3; x�n3+1; : : : ; x; w; f(w); : : : ; fn1(w)gcannot be �-shadowed by any true trajectory. Indeed, same argumentscan prove that there exists > 0, such that every �-pseudo trajectorythat comes within of �x�n3 cannot be �-shadowed by any true trajec-tory. Without loss of generality, we assume � is su�ciently small. Thenfrom Proposition 2.1, almost every �-pseudo trajectory comes within of �x�n3 . Thus we complete the proof. 2Theorem 2.2 is our main result. Note that dimension variability isthe key. 3. ExamplesAbraham and Smale [21] give an example of an open set of maps ona four dimensional manifold with the following property. There are twodisjoint compact invariant sets, �1 (a 2-torus on which the map is theThom di�eomorphism de�ned by the linear isomorphism with matrix0B@ 1 21 1 1CA

14 GUO-CHENG YUAN AND JAMES A. YORKE) and �2 (a saddle �xed point whose stable manifold is one dimen-sional), and there are �nite number of trajectories asymptotic to �1 asn ! �1 and to �2 as n ! 1. For a dense set of maps the negativelimit sets of these trajectories are each equal to �1 and for another denseset at least one of these trajectories has a negative limit set that is aperiodic orbit, a proper subset of �1. While they emphasize -stabilityand do not mention shadowability, it is clear that such trajectories can-not be shadowed by trajectories in nearby systems. It can be shownthat the map satis�es our k-cone condition with k = 1 at least for somechoices of the parameter. We could adapt Theorem 2.2 as well to thisnon-attracting case and show the existence of a nonshadowable pseudotrajectory.In this section we present two other examples. The goal is to showthat there exists an open set of maps for which every point is absolutelynonshadowable.Example 3.1. Kostelich et al. [22] have considered a map on T 2 givenby: f(y; z) = (2y mod 2�; (y + z) + c sin(y + z) mod 2�) (3.1)where c is a parameter in a neighborhood of 0:6. The Jacobian matrixof the map is:Df(y;z) = 0B@ 2 01 + c cos(y + z) 1 + c cos(y + z) 1CAKostelich et al. [22] have proved that f has a dense trajectory, whichimplies that T 2 is an attractor as de�ned in Section 2; also, T 2 continuesto have a dense trajectory under small Cr-perturbations, where r >

NONSHADOWABLE SYSTEMS 151. There are two �xed points of f : (0,0), a source, and (0, �), asaddle. These �xed points are hyperbolic and persist under small Cr-perturbations. Therefore, there exists an open set of maps which havedimension variability.In contrast, no 2-dimensional di�eomorphism may have dimensionvariability, because the !-limit set of a trajectory does not contain asource unless the trajectory is the source itself.By Theorem 2.2, in order to show nonshadowability it su�ces toshow that there exists a positively invariant 1-cone �eld. The existencefollows from the fact that the y-coordinate expands by a factor of 2,more than the expansion of the z-coordinate. Speci�cally, it can be seenas follows. Let M be a constant which is greater than (1 + c)=(1� c),A tangent vector (1; t) at (y; z) maps to (2; (1+ t)(1+ c cos(y+ z))) atf(y; z). If jtj �M , then j(1+t)(1+c cos(y+z))j � (1+c)(1+jtj) < 2M .Therefore f(u; v) : jvj � M jujg is a positively invariant 1-cone �eldfor f . Indeed, it continues to be positively invariant under small Cr-perturbations. Thus there exists an open set of maps for which everypoint is absolutely nonshadowable.We remark that it is also easy to make a three dimensional examplein which there is a two dimensional attracting torus which has thedescribed dynamics. This can be done by crossing the above examplewith a circle on which the map has an attracting �xed point.Example 3.2. The torus map in Example 3.1 is not one-to-one. How-ever, we can use a technique introduced by Smale [23] to build a di�eo-morphism example from an endomorphism one. The idea is to replacethe circle map (the y-component of Equation (3.1)) by a \solenoid"di�eomorphism on a solid torus S1 � D2. Let M = S3� S1, and letM = S1 � D2 � S1. M is a submanifold (with boundary) of M . On

16 GUO-CHENG YUAN AND JAMES A. YORKEM we can de�ne coordinates (t; z; s) such that t 2 S1, z 2 D2 (z isa complex number), and s 2 S1. De�ne �g : M ! M (see [24], pages294-298) as (t; z; s) 7! (2t; 14z + 12e2�ti; (t + s) + c sin(t + s)). �g can beextended to a global di�eomorphism g : M ! M such that gjM = �g.It follows immediately that there is an attractor A � S1 � D2 � S1,which is the Cartesian product of a solenoid attractor in S1 �D2 andS1. Using similar arguments to those in Kostelich et al. [22], we canprove that A continues to be an attractor. Similarly to Example 3.1,the attractor A had dimension variability and a positively invariant1-cone �eld. Therefore Theorem 2.2 implies that every point in thebasin of A is absolutely nonshadowable. Indeed, �g can be extended insuch a way that A attracts almost every trajectory in S3 � S1. Thusthere exists an open set of di�eomorphisms for which every point isabsolutely nonshadowable.We thank Leny Nusse, Lan Wen and Vadim Kaloshin for helpfulcommunications. References[1] D. V. Anosov. Geodesic ows on closed riemann manifolds with negative cur-vature. Proceedings of the Steklov Institute of Mathematics, 90, 1967.[2] Rufus Bowen. !-limit sets for Axiom A di�eomorphisms. Journal of Di�erentialEquations, 18(2):333{339, 1975.[3] Michael Benedicks and Lennart Carleson. The dynamics of the H�enon map.Annals of Mathematics(2), 133(1):73{169, 1991.[4] P. J. Myrberg. Sur l'it�eration des polynomes r�eels quadratiques. Journal deMath�ematiques Pures et Appliqu�ees (9), 41:339{351, 1962.[5] Ethan M. Coven, Ittai Kan, and James A. Yorke. Pseudo-orbit shadowing inthe family of tent maps. Transactions of the American Mathematical Society,308(1):227{241, 1988.

NONSHADOWABLE SYSTEMS 17[6] Stephen Smale and Robert F. Williams. The qualitative analysis of a di�er-ence equation of population growth. Journal of Mathematical Biology, 3(1):1{4,1976.[7] Stephen M. Hammel, James A. Yorke, and Celso Grebogi. Do numerical orbitsof chaotic dynamical processes represent true orbits? Journal of Complexity,3(2):136{145, 1987.[8] Stephen M. Hammel, James A. Yorke, and Celso Grebogi. Numerical orbitsof chaotic processes represent true orbits. Bulletin of American MathematicalSociety, 19(2):465{469, 1988.[9] Tim Sauer and James. A. Yorke. Rigorous veri�cation of trajectories for thecomputer simulation of dynamical systems. Nonlinearity, 4(2):961{979, 1991.[10] Brian A. Coomes, H�useyin Ko�cak, and Kenneth J. Palmer. Periodic shadowing.In Chaotic Numerics, Contemporary Mathematics Series 172., pages 115{130.American Mathematical Society, Providence, Rhode Island, 1994.[11] Brian A. Coomes, H�useyin Ko�cak, and Kenneth J. Palmer. Shadowing in dis-cete dynamical systems. In Six Lectures on Dynamical Systems, pages 163{211.World Scienti�c Publishing, River Edge, New Jersey, 1996.[12] Shui-Nee Chow and Kenneth J. Palmer. On the numerical computation oforbits of dynamical systems: the one-dimensional case. Journal of Dynamicsand Di�erential Equations, 3(3):361{379, 1991.[13] Shui-Nee Chow and Kenneth J. Palmer. On the numerical computation of or-bits of dynamical systems: the higher-dimensional case. Journal of Complexity,8(4):398{423, 1992.[14] Tim Sauer, Celso Grebogi, and James A. Yorke. How long do numerical chaoticsolutions remain valid? Physical Review Letters, 79:59, 1997.[15] Leon Poon, Celso Grebogi, Tim Sauer, and James A. Yorke. Limits to deter-ministic modeling. Preprint.[16] William Feller. An introduction to probability theory and its applications. JohnWiley & Sons, Inc., 1957.[17] Lai-Sang Young. Stochastic stability of hyperbolic attractors. Ergodic Theoryand Dynamical Systems, 6:311{319, 1986.[18] Yuri Kifer. Random perturbations of dynamical systems. Birkhauser, 1988.

18 GUO-CHENG YUAN AND JAMES A. YORKE[19] Jacques Hadamard. Sur l'it�eration et les solutions asymptotiques des �equationsdi��erentielles. Bulletin de la Soci�et�e Math�ematique de France, 29:224{228,1901.[20] Anatole Katok and Boris Hasselblatt. Introduction to modern theory of dy-namical systems. Cambridge University Press, 1995.[21] Ralph H. Abraham and Stephen Smale. Nongenericity of -stability. In Globalanalysis (Proceedings of Symposia in Pure Mathemaitcs, vol. 14, 1968), vol-ume 14, pages 5{8. American Mathematical Society, 1970.[22] Eric J. Kostelich, Ittai Kan, Celso Grebogi, Edward Ott, and James A. Yorke.Unstable dimension variability: A source of nonhyperbolicity in chaotic sys-tems. Preprint.[23] Stephen Smale. Di�erentiable dynamical systems. Bulletin of the AmericanMathematical Society, 73:747{817, 1967.[24] Clark Robinson. Dynamical systems : stability, symbolic dynamics, and chaos.CRC Press, Inc., 1995.Institute for Physical Science and Technology, and Department ofMathematics, University of Maryland, College Park, MD 20742, USAE-mail address : [email protected], [email protected]